At the top boundary we enforce a constant potential temperature gradient, zero vertical velocity, and zero shear stress boundary condition. At the bottom boundary, we enforce a zero potential temperature flux and the classical wall stress formulation [1]}, [2]},
\(\tau _{xz} = - \left( \frac{\kappa }{\ln z_1/z_0} \right)^2 \left( \overline{\widetilde{u}}^2 +\overline{\widetilde{v}}^2 \right)^{ 1/2 } \overline{\widetilde{u}}, \quad \tau _{yz} = - \left( \frac{\kappa }{\ln z_1/z_0} \right)^2 \left( \overline{\widetilde{u}}^2 + \overline{\widetilde{v}}^2 \right)^{ 1/2 } \overline{\widetilde{v}},\) 
